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Using ab initio calculations based on the correlated band theory, we haveinvestigated the quasi-one-dimensional chain system $\beta$-V$_2$O(PO$_4$),showing both charge and spin orderings. Even in the uncorrelated region, thepure transition from the tetragonal to the monoclinic structure leads to asizable charge difference between the two types of V ions, regardless ofmagnetic orders. In the ferrimagnetic phase, inclusion of the on-site Coulombrepulsion $U$ leads to a full orbital-polarization of V1 ($t_{2g}^{3\uparrow}$,$S=\frac{3}{2}$) and V2 ($a_{1g}^{1\downarrow}e_g^{\prime{1\downarrow}}$,$S=1$) above $U^c_{eff}\approx3.5$ eV, leading to local spin moments of 2.30and --1.54 $\mu_B$, respectively, with small orbital moments of severalhundredth $\mu_B$. So, the net moment is nearly 1 $\mu_B$ per formula unit,which is about 2--3 times larger than the experimental value. Our results showsignificant variations, strongly depending on the strength of $U_{eff}$, inenergy differences between various magnetic states as well as a small magneticanisotropy. These results suggest that the substantial difference between thecalculated and experimental moments is attributed to quantum fluctuation of thepyrochlore-like weakly linked V$_4$ tetrahedral structure. Our findings areexpected to provide a good platform to investigate the interplay among thecharge-, spin-, and lattice-degrees of freedom, and geometrical frustration.

Charge and Orbital Orderings, and Frustration in Quasi-one-dimensional Ferrimagnetic Insulator $\beta$-V$_2$O(PO$_4$)